We started discussing recursive sequences in my Analysis I class last week (plus related questions 'is it monotonic?', 'does it converge?', 'is it bounded?' etc.). In my homework set I have the following problems:
(1) Show that the sequence given by
$$ \begin{aligned} a_1=2 \quad \quad \quad \quad \ \\ a_n=\frac{1}{2} \left(a_{n-1} + 6 \right) \end{aligned} $$
Is increasing and bounded above
And
(2) Consider the recursive sequence $a_{n+1} = 7-\frac{10}{a_n}$, with initial datum $a_1 = 4$. Compute the first two or three values. Show that it is bounded by 2 and 5, that it is increasing, and compute the limit.
During class, we discussed that the way to find whether a recursive sequence is increasing/decreasing is to compute the difference of $a_{n+1} - a_n$.
One of my classmates explained to me that this difference for (1) equals $3-\frac{a_n}{2}$, but he couldn't explain to me how he got to this value.
Concretely, my question is:
a) How do I compute the difference between $a_{n-1} - a_n$? What sort of reasoning should I employ to do this? To be honest, the subscripts mess me up quite a bit.
b) I had some trouble finding literature on this. Do you know of any resources where I can read up on this?
(Note: I copied the questions here for completeness, but I'm not asking you to help me to show that the sequences is bounded. I can reason through the induction proof, I'm just a bit clueless about this first step.)